H. Mawi, C. B. Ndiaye
In this note, we show that a natural optimal control problem for the $\infty$-obstacle problem admits an optimal control which is also an optimal state. Moreover, we show the convergence of the minimal value of an optimal control problem for the $p$-obstacle problem to the minimal value of our optimal control problem for the $\infty$-obstacle problem, as $p\to\infty$.
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This paper deals with the obstacle problem for the infinity Laplacian. The main results are a characterization of the solution through comparison with cones that lie above the obstacle and the sharp $C^{1,1/3}$--regularity at the free boundary.
Steady vortex-patch flows with circulation past an obstacle in an infinity long strip
本文研究了无限长条带中绕过障碍物的稳态Euler流，基于新Green函数结构、Kirchhoff-Routh函数极小值点及动能极大值的存在性，分析了障碍物、无穷远速度和环流对涡斑流动存在性的影响。
In this paper, we consider the thin obstacle problem in $\mathbb{R}^2$ with data at infinity. We first prove the existence and uniqueness of it. Then we show that its symmetric solutions are actually half-space solutions. Our results are needed when classifying the half-space $(2k-\frac{1}{2})$-homogeneous solutions to the thin obstacle problems in $\mathbb{R}^3$. It is a generalization of one part of Savin-Yu's work \cite{savin2021halfspace} on classifying the half-space $\frac{7}{2}$-homogeneous solutions.
We consider the obstacle problem for the infinity Laplace equation. Given a Lipschitz boundary function and a Lipschitz obstacle we prove the existence and uniqueness of a super infinity-harmonic function constrained to lie above the obstacle which is infinity harmonic where it lies strictly above the obstacle. Moreover, we show that this function is the limit of value functions of a game we call obstacle tug-of-war.
This paper provides asymptotic structure at spatial infinity of plane steady Stokes flow in exterior domains when the obstacle is rotating with constant angular velocity. The result shows that there is no longer Stokes paradox due to the rotating effect.
Global solutions to the thin obstacle problem with superquadratic growth
本文研究了具有超二次增长的薄障碍问题全局解的刚性/柔性，根据无穷远展开对有界正集解进行分类，并证明有界接触集解可逼近任意紧集，揭示了经典障碍问题中不存在的现象。
Fractional Infinity Laplacian with Obstacle
本文研究了带有障碍的分数无穷 Laplace 算子问题，证明了在特定条件下该问题存在解，并给出了解的 Hölder 连续性。
Optimality in nonlocal time-dependent obstacle problems
本文研究非局部时变障碍问题中quasiconvex性对时间导数正则性和连续性条件的影响。
Compact contact sets of sub-quadratic solutions to the thin obstacle problem
本文研究无穷远处至多二次增长的薄障碍问题全局解，证明每个椭球都可作为某解的接触集，且具有紧致接触集的解其接触集必为椭球。
In this paper, we study the problem concerning the approximation of a rigid obstacle for flows governed by the stationary Navier-Stokes equations in the two-dimensional case. The idea is to consider a highly viscous fluid in the place of the obstacle. Formally, as the fluid viscosity goes to infinity inside the region occupied by the obstacle, we obtain the original problem in the limit. The main goal is to establish a better regularity of approximate solutions. In particular, the pointwise estimate for the gradient of the velocity is proved. We give numerical evidence that the penalized solut…
Optimal Hölder regularity for solutions to Signorini-type obstacle problems
本文研究一类障碍问题弱解的存在性、唯一性和正则性，建立了Signorini型问题（障碍条件施加在余维1子集上）的最优Hölder正则性，并应用于不规则障碍的经典障碍问题。
We consider an optimal rearrangement maximization problem involving the fractional Laplace operator $(-Δ)^s$, $0<s<1$, and the Gagliardo-Nirenberg seminorm $[u]_s$. We prove the existence of a maximizer, analyze its properties and show that it satisfies the unstable fractional obstacle problem equation for some $α>0$ $$(-Δ)^s u=χ_{\{u>α\}}.$$
In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson \cite{An16} and the epiperimetric inequality from \cite{FS16}, \cite{GPSVG15}, we prove the optimal $C^{1,\min\{α,1/2\}}$ regularity of solutions in the presence of $C^{0,α}$ coefficients $a^{ij}$ and $C^{1,α}$ obstacles $ϕ$. Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $C^{1,γ}$ manifold for some $γ\in (0,1)$.
A necessary condition for extremality of solutions to autonomous obstacle problems with general growth
该研究针对满足凸性与无穷远超线性条件的自治障碍问题，利用原始-对偶表述刻画其存在且唯一的解，证明采用凸分析经典论证与变分法技巧，给出了解的极值性的必要条件。
We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take int…
This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.
Asymptotics as $s\searrow 0$ of the nonlocal nonparametric Plateau problem with obstacles
本文研究非局部极小图在障碍下的存在性、先验估计及两种设定的等价性，发现当分数参数趋小时，非局部极小图会完全贴合障碍并渐近离开域内。
This paper is interested in the problem of optimal stopping in a mean field game context. The notion of mixed solution is introduced to solve the system of partial differential equations which models this kind of problem. This notion emphasizes the fact that Nash equilibria of the game are in mixed strategies. Existence and uniqueness of such solutions are proved under general assumptions for both stationary and evolutive problems.
ACF Almost Monotonicity at Infinity with Applications to Perturbed Global Solutions
针对经典障碍问题的扰动全局解，利用 Alt-Caffarelli-Friedman 泛函的大尺度几乎单调性公式，证明在足够远的正则点附近，零集的横截面是椭球的 C^2 扰动。
In this paper, the well-posedness and optimal convergence rates of subsonic irrotational flows through a three dimensional infinitely long nozzle with a smooth obstacle inside are established. More precisely, the global existence and uniqueness of the uniformly subsonic flow are obtained via variational formulation as long as the incoming mass flux is less than a critical value. Furthermore, with the aid of delicate choice of weight functions, we prove the optimal convergence rates of the flow at far fields via weighted energy estimates and Nash-Moser iteration.
We consider an optimal rearrangement minimization problem involving the fractional Laplace operator $(-Δ)^s$, $0<s<1$, and Gagliardo-Nirenberg seminorm $|u|_s$. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies $$ -(-Δ)^s U-χ_{\{U\leq 0\}}\min\{-(-Δ)^s U^+;1\}=χ_{\{U>0\}}, $$ which happens to be the fractional analogue of the normalized obstacle problem $Δu=χ_{\{u>0\}}$. A new section analyzing $s \to 1$ has been added.
We study the parabolic free boundary problem of obstacle type $$ \lap u-\frac{\partial u}{\partial t}= fχ_{u\ne 0}. $$ Under the condition that $f=Hv$ for some function $v$ with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution $u$. Both the regularity and the assumptions are optimal. Using this result and assuming that $f$ is Dini continuous, we prove that the free boundary is, near so called low energy points, a $C^1$ graph. Our result completes the theory for this type of problems for the heat operator.
Optimal regularity for the fully nonlinear thin obstacle problem
本文研究了完全非线性薄障碍问题解的最优正则性，通过证明正则点处爆破的唯一性及解的展开，确立了最优指数 $α_F$ 使得解在障碍两侧均为 $C^{1,α_F}$，并证明了旋转不变算子下解始终为 $C^{1,1/2}$。
We study the double-obstacle problem for the p-Laplace operator, p 2 [2;1). We prove that for Lipschitz boundary data and Lipschitz obstacles, viscosity solutions are unique and coincide with variational solutions. They are also uniform limits of solutions to discrete min-max problems that can be interpreted as the dynamic programming principle for appropriate tug-ofwar games with noise. In these games, both players in addition to choosing their strategies, are also allowed to choose stopping times. The solutions to the double-obstacle problems are limits of values of these games, when the ste…
In this work, we investigate a distributed optimal control problem for an extended phase field system of Cahn--Hilliard type which physical context is that of tumor growth dynamics. In a previous contribution, the author has already studied the corresponding problem for the logarithmic potential. Here, we try to extend the analysis by taking into account a non-smooth singular nonlinearity, namely the double obstacle potential. Due to its non-smoothness behavior, the standard procedure to characterize the necessary conditions for the optimality cannot be performed. Therefore, we follow a differ…
Finding the convex hull of a set using the flow by minimal curvature with an obstacle. A game theoretical approach
本文研究带障碍的最小主曲率几何流，通过水平集障碍问题的博弈论近似，证明正超水平集随时间收敛到障碍的凸包。
We consider a new type of obstacle problem in the cylindrical domain $Ω=D\times (0,1)$ arising from minimization of the functional $$ \int_Ω\frac{1}{2}|\nabla u|^2+χ_{\{v>0\}}udx, $$ where $v(x')=\int_0^1 u(x', t) dt $. We prove several existence and regularity results and show that the comparison principle does not hold for minimizers. This problem is derived from a classical optimal rearrangement problem in a cylindrical domain, under the constraint that the force function does not depend on the $x_n$ variable of the cylindrical axis. A mistake in the Theorem 4.2 of the previous version has …
Global solutions to the fully nonlinear obstacle problem
本文研究完全非线性障碍问题的全局解，证明其重合集的四类类型，并建立紧重合集全局解与无穷远渐近行为函数族间的双射关系。
In this paper we prove the optimal $C^{1,1}(B_\frac12)$-regularity for a general obstacle type problem $$ \lap u = fχ_{\{u\neq 0\}}\textup{in $B_1$}, $$ under the assumption that $f*N$ is $C^{1,1}(B_1)$, where $N$ is the Newtonian potential. This is the weakest assumption for which one can hope to get $C^{1,1}$-regularity. As a by-product of the $C^{1,1}$-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point $x^0$, the free boundary is locally a $C^1$-graph close to $x^0$, provided $f$ is Dini. This completely settles the que…
This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution in a low regularity set-up. We first discuss the case of zero obstacle and $W^{1,p}$ metrics with $p\in(n+1,\infty]$. In this framework, we prove the $C^{1,α}$ regularity of the regular free boundary and derive the leading order asymptotic expansion of solutions at regular free boundary points. We further show the optimal $C^{1,\min\{1-\frac{n+1}{p}, \frac{1…
In this paper we discuss the obstacle problem for the $p$-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the $p$-obstacle problem: Find the smallest $u$ such that $$ \hbox{div} (|\nabla u|^{p-2}\nabla u) \leq 0, \qquad u\geq ϕ, \qquad \hbox{in } B_1, $$ with $ϕ\in C^{1,1}(B_1)$ and given boundary datum on $\partial B_1$. We prove that the solution is uniformly $C^{1,1}$ at free boundary points. Similar results are obtained in the c…
Building upon the recent results in \cite{FoSp17} we provide a thorough description of the free boundary for the fractional obstacle problem in $\mathbb{R}^{n+1}$ with obstacle function $φ$ (suitably smooth and decaying fast at infinity) up to sets of null $\mathcal{H}^{n-1}$ measure. In particular, if $φ$ is analytic, the problem reduces to the zero obstacle case dealt with in \cite{FoSp17} and therefore we retrieve the same results: (i) local finiteness of the $(n-1)$-dimensional Minkowski content of the free boundary (and thus of its Hausdorff measure), (ii) $\mathcal{H}^{n-1}$-rectifiabili…
We consider various versions of the obstacle and thin-obstacle problems, we interpret them as variational inequalities, with non-smooth constraint, and prove that they satisfy a new constrained Lojasiewicz inequality. The difficulty lies in the fact that, since the constraint is non-analytic, the pioneering method of L. Simon (Ann. of Math. 118(3), 1983) does not apply and we have to exploit a better understanding on the constraint itself. We then apply this inequality to two associated problems. First we combine it with an abstract result on parabolic variational inequalities, to prove the co…
This note is a synthesis of my reflexions on some questions that have emerged during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type" concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form L[u](x) + f (u(x)) = 0, for x $\in$ R n \ K, where K $\subset$ R N is a bounded smooth compact "obstacle", L is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator L is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymp…
We prove optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.
We consider the semilinear problem \[ Δu = λ_+ \left(-\log u^+\right) 1_{\{u > 0\}} - λ_- \left(-\log u^- \right) 1_{\{u < 0\}} \qquad \hbox{ in } B_1, \] where $B_1$ is the unit ball in $\mathbb{R}^n$ and assume $λ_+, λ_- > 0$. Using a monotonicity formula argument, we prove an optimal regularity result for solutions: $\nabla u$ is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one alrea…
Optimal regularity of the thin obstacle problem by an epiperimetric inequality
本文利用负能量 $W_\frac32$ 的超周长不等式排除 $\lambda\in(1,\frac32)$ 的齐次全局解，以频率间隙给出薄障碍问题最优 $C^{1,\frac12}$ 正则性的新证明。
We study certain obstacle type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.
We prove optimal regularity for the double obstacle problem when obstacles are given by solutions to Hamilton-Jacobi equations that are not $C^2$. When the Hamilton-Jacobi equation is not $C^2$ then the standard Bernstein technique fails and we loose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speed in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that $C^1$-solutions to the Hamilton-Jacobi equation $$ \pm |\nabla h-a(x)|^2=\pm …
We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an $n$-dimensional Levy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependenc…
We prove the optimal $W^{2, \infty }$ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be $C^1$ or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal $W^{2, \infty }$ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the sec…
Optimal Asymptotic Behavior at Infinity of Ancient Solution to the Parabolic Monge-Ampère Equation with Slow Perturbation Term
研究抛物型Monge-Ampère方程在慢扰动项下的古解在无穷远处的最优渐进行为，扩展了椭圆估计到抛物情形。
Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity -- sometimes called an Oseen condition. By a suitable change of coordinates the problem is transformed to an non-autonomous problem with unbounded drift terms on a fixed exterior domain $Ω\subset \R^d$. It is shown that the solution to the linearized problem is governed by a strongly continuous evolution system $\{T_Ω(t,s)\}_{t\geq s\geq0}$ on $L^p_σ(Ω)$ for $1<p<\infty$. Moreover, $L^p$-$L^q$ smoothing properties and gradient estimates of $…
We analyse the obstacle problem for the nonlocal parabolic operator \[\partial_t u + (-Δ)^{s} u - b \cdot \nabla u - \mathcal{I}u - ru,\] where $b\in\mathbb{R}^n$, $r\in\mathbb{R}$, and $\mathcal{I}$ is a nonlocal lower order diffusion operator with respect to the fractional Laplace operator $(-Δ)^{s}$. This model appears in the study of American options pricing when the stochastic process governing the stock price is assumed to be a purely jump process. We study the existence and the uniqueness of solutions to the obstacle problem, and we prove optimal regularity of solutions in space, and al…
We establish the optimal $C_{H}^{1,1}$ interior regularity of solutions to \[ Δ_{H}u=fχ_{\{u\ne0\}}, \] where $Δ_{H}$ denotes the sub-Laplacian operator in a stratified group. We assume the weakest regularity condition on $f$, namely $f*Γ$ is $C_{H}^{1,1}$, where $Γ$ is the fundamental solution of $Δ_{H}$. The $C_{H}^{1,1}$ regularity is understood in the sense of Folland and Stein. In the classical Euclidean setting, the first seeds of the above problem are already present in the 1991 paper of Sakai and are also related to quadrature domains. As a special instance of our results, when $u$ is …
Higher order interpolative geometries and gradient regularity in evolutionary obstacle problems
本文针对退化情形 $p>2$ 的演化 $p$-Laplace 型障碍问题，在无时间导数假设下，利用高阶内蕴插值几何证明了自由边界处及跨自由边界的最优 $C^{1,α}$ 梯度正则性，并给出将其扩展至整个柱体的内蕴准则。
In a wide class of the so called Obstacle Problems of parabolic type it is shown how to improve the optimal regularity of the solution and as a consequence how to obtain space-time regularity of the corresponding free boundary.
Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount $ε>0$ (as is done in the classical case), it is a $s$-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call "infinity fractional Laplacian". W…
In this paper we establish the optimal interior regularity and the $C^{1,γ}$ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.
In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.
In this paper we are concerned with a two-penalty boundary obstacle problem of interest in thermics, fluid dynamics and electricity. Specifically, we prove existence, uniqueness and optimal regularity of the solutions, and we establish structural properties of the free boundary.
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory with some proofs: from the optimal regularity of solutions and classification of free boundary points, to more recent results on the non-regular part of the free boundary and generic regularity.
We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity $C^{1,1/2}$. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary $C^{1,β}$ hypersurface, $β>1/2$, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren's monotonicity formula and the optimal regularity of global solutions.
Under a sharp asymptotic growth condition at infinity, we prove a Liouville type theorem for the inhomogeneous porous medium equation, provided it stays universally close to the heat equation. Additionally, for the homogeneous equation, we show that for the conclusion to hold, it is enough to assume the sharp asymptotic growth at infinity only in the space variable. The results are optimal, meaning that the growth condition at infinity cannot be weakened.
We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games with a variational structure. We prove L infinity regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent's trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L infinity results have already been found by P.-L. Lions in his course on Mean Field Games, using a proof based on …
In this paper, we study a natural optimal control problem associated to the Paneitz obstacle problem on closed 4-dimensional Riemannian manifolds. We show the existence of an optimal control which is an optimal state and induces also a conformal metric with prescribed Q-curvature. We show also C^{\infty}-regularity of optimal controls and some compactness results for the optimal controls. In the case of the 4-dimensional standard sphere, we characterize all optimal controls.
Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains
研究外域上超临界拉格朗日均曲率方程解在无穷远处的渐进行为，采用尺度依赖的差商方法和非局部势方法，无需对f二次求导，仅在Lipschitz正则性下获得定量Hessian收敛，得到所有维度和所有衰减率β>0的最优渐进行为，包括临界对数情况。
We consider strictly log-concave measures, whose bounds degenerate at infinity. We prove that the optimal transport map from the Gaussian onto such a measure is locally Lipschitz, and that the eigenvalues of its Jacobian have controlled growth at infinity.
Systems of Fully Nonlinear Degenerate Elliptic Obstacle problems with Dirichlet boundary conditions
本文研究带Dirichlet边界条件的完全非线性退化椭圆障碍问题系统，证明了含互联障碍极小化问题对应完全非线性椭圆系统粘性解的存在唯一性，该系统可应用于有界域最优切换问题等领域。
In this paper, we consider the analogous of the obtacle problem in $H_0^1(Ω)$, on the space $W^{1,p}_0(Ω)$. We prove an existence and uniqueness of the result. In a second time, we define the optimal control problem associated. The results, here enclosed, generalize the one obtained by D.R. Adams, S. Lenhard
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